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Theorem tgbtwnconn1lem1 25688
Description: Lemma for tgbtwnconn1 25691. (Contributed by Thierry Arnoux, 30-Apr-2019.)
Hypotheses
Ref Expression
tgbtwnconn1.p 𝑃 = (Base‘𝐺)
tgbtwnconn1.i 𝐼 = (Itv‘𝐺)
tgbtwnconn1.g (𝜑𝐺 ∈ TarskiG)
tgbtwnconn1.a (𝜑𝐴𝑃)
tgbtwnconn1.b (𝜑𝐵𝑃)
tgbtwnconn1.c (𝜑𝐶𝑃)
tgbtwnconn1.d (𝜑𝐷𝑃)
tgbtwnconn1.1 (𝜑𝐴𝐵)
tgbtwnconn1.2 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
tgbtwnconn1.3 (𝜑𝐵 ∈ (𝐴𝐼𝐷))
tgbtwnconn1.m = (dist‘𝐺)
tgbtwnconn1.e (𝜑𝐸𝑃)
tgbtwnconn1.f (𝜑𝐹𝑃)
tgbtwnconn1.h (𝜑𝐻𝑃)
tgbtwnconn1.j (𝜑𝐽𝑃)
tgbtwnconn1.4 (𝜑𝐷 ∈ (𝐴𝐼𝐸))
tgbtwnconn1.5 (𝜑𝐶 ∈ (𝐴𝐼𝐹))
tgbtwnconn1.6 (𝜑𝐸 ∈ (𝐴𝐼𝐻))
tgbtwnconn1.7 (𝜑𝐹 ∈ (𝐴𝐼𝐽))
tgbtwnconn1.8 (𝜑 → (𝐸 𝐷) = (𝐶 𝐷))
tgbtwnconn1.9 (𝜑 → (𝐶 𝐹) = (𝐶 𝐷))
tgbtwnconn1.10 (𝜑 → (𝐸 𝐻) = (𝐵 𝐶))
tgbtwnconn1.11 (𝜑 → (𝐹 𝐽) = (𝐵 𝐷))
Assertion
Ref Expression
tgbtwnconn1lem1 (𝜑𝐻 = 𝐽)

Proof of Theorem tgbtwnconn1lem1
StepHypRef Expression
1 tgbtwnconn1.p . 2 𝑃 = (Base‘𝐺)
2 tgbtwnconn1.m . 2 = (dist‘𝐺)
3 tgbtwnconn1.i . 2 𝐼 = (Itv‘𝐺)
4 tgbtwnconn1.g . 2 (𝜑𝐺 ∈ TarskiG)
5 tgbtwnconn1.b . 2 (𝜑𝐵𝑃)
6 tgbtwnconn1.j . 2 (𝜑𝐽𝑃)
7 tgbtwnconn1.a . 2 (𝜑𝐴𝑃)
8 tgbtwnconn1.h . 2 (𝜑𝐻𝑃)
9 tgbtwnconn1.1 . 2 (𝜑𝐴𝐵)
10 tgbtwnconn1.e . . 3 (𝜑𝐸𝑃)
11 tgbtwnconn1.d . . . 4 (𝜑𝐷𝑃)
12 tgbtwnconn1.3 . . . 4 (𝜑𝐵 ∈ (𝐴𝐼𝐷))
13 tgbtwnconn1.4 . . . 4 (𝜑𝐷 ∈ (𝐴𝐼𝐸))
141, 2, 3, 4, 7, 5, 11, 10, 12, 13tgbtwnexch 25614 . . 3 (𝜑𝐵 ∈ (𝐴𝐼𝐸))
15 tgbtwnconn1.6 . . 3 (𝜑𝐸 ∈ (𝐴𝐼𝐻))
161, 2, 3, 4, 7, 5, 10, 8, 14, 15tgbtwnexch 25614 . 2 (𝜑𝐵 ∈ (𝐴𝐼𝐻))
17 tgbtwnconn1.f . . 3 (𝜑𝐹𝑃)
18 tgbtwnconn1.c . . . 4 (𝜑𝐶𝑃)
19 tgbtwnconn1.2 . . . 4 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
20 tgbtwnconn1.5 . . . 4 (𝜑𝐶 ∈ (𝐴𝐼𝐹))
211, 2, 3, 4, 7, 5, 18, 17, 19, 20tgbtwnexch 25614 . . 3 (𝜑𝐵 ∈ (𝐴𝐼𝐹))
22 tgbtwnconn1.7 . . 3 (𝜑𝐹 ∈ (𝐴𝐼𝐽))
231, 2, 3, 4, 7, 5, 17, 6, 21, 22tgbtwnexch 25614 . 2 (𝜑𝐵 ∈ (𝐴𝐼𝐽))
241, 2, 3, 4, 7, 5, 10, 8, 14, 15tgbtwnexch3 25610 . . 3 (𝜑𝐸 ∈ (𝐵𝐼𝐻))
251, 2, 3, 4, 7, 18, 17, 6, 20, 22tgbtwnexch 25614 . . . . 5 (𝜑𝐶 ∈ (𝐴𝐼𝐽))
261, 2, 3, 4, 7, 5, 18, 6, 19, 25tgbtwnexch3 25610 . . . 4 (𝜑𝐶 ∈ (𝐵𝐼𝐽))
271, 2, 3, 4, 5, 18, 6, 26tgbtwncom 25604 . . 3 (𝜑𝐶 ∈ (𝐽𝐼𝐵))
281, 2, 3, 4, 7, 5, 11, 10, 12, 13tgbtwnexch3 25610 . . . 4 (𝜑𝐷 ∈ (𝐵𝐼𝐸))
291, 2, 3, 4, 7, 18, 17, 6, 20, 22tgbtwnexch3 25610 . . . . 5 (𝜑𝐹 ∈ (𝐶𝐼𝐽))
301, 2, 3, 4, 18, 17, 6, 29tgbtwncom 25604 . . . 4 (𝜑𝐹 ∈ (𝐽𝐼𝐶))
311, 2, 3, 4, 6, 17axtgcgrrflx 25582 . . . . 5 (𝜑 → (𝐽 𝐹) = (𝐹 𝐽))
32 tgbtwnconn1.11 . . . . 5 (𝜑 → (𝐹 𝐽) = (𝐵 𝐷))
3331, 32eqtr2d 2796 . . . 4 (𝜑 → (𝐵 𝐷) = (𝐽 𝐹))
34 tgbtwnconn1.8 . . . . . 6 (𝜑 → (𝐸 𝐷) = (𝐶 𝐷))
35 tgbtwnconn1.9 . . . . . 6 (𝜑 → (𝐶 𝐹) = (𝐶 𝐷))
3634, 35eqtr4d 2798 . . . . 5 (𝜑 → (𝐸 𝐷) = (𝐶 𝐹))
371, 2, 3, 4, 10, 11, 18, 17, 36tgcgrcomlr 25596 . . . 4 (𝜑 → (𝐷 𝐸) = (𝐹 𝐶))
381, 2, 3, 4, 5, 11, 10, 6, 17, 18, 28, 30, 33, 37tgcgrextend 25601 . . 3 (𝜑 → (𝐵 𝐸) = (𝐽 𝐶))
39 tgbtwnconn1.10 . . . 4 (𝜑 → (𝐸 𝐻) = (𝐵 𝐶))
401, 2, 3, 4, 10, 8, 5, 18, 39tgcgrcomr 25594 . . 3 (𝜑 → (𝐸 𝐻) = (𝐶 𝐵))
411, 2, 3, 4, 5, 10, 8, 6, 18, 5, 24, 27, 38, 40tgcgrextend 25601 . 2 (𝜑 → (𝐵 𝐻) = (𝐽 𝐵))
421, 2, 3, 4, 5, 6axtgcgrrflx 25582 . 2 (𝜑 → (𝐵 𝐽) = (𝐽 𝐵))
431, 2, 3, 4, 5, 6, 5, 7, 8, 6, 9, 16, 23, 41, 42tgsegconeq 25602 1 (𝜑𝐻 = 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2140  wne 2933  cfv 6050  (class class class)co 6815  Basecbs 16080  distcds 16173  TarskiGcstrkg 25550  Itvcitv 25556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-nul 4942
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-iota 6013  df-fv 6058  df-ov 6818  df-trkgc 25568  df-trkgb 25569  df-trkgcb 25570  df-trkg 25573
This theorem is referenced by:  tgbtwnconn1lem2  25689  tgbtwnconn1lem3  25690
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